On the set \(\mathcal {U}=\mathbb {H}\) . The Karush-Kuhn-Tucker (KKT) theory for infinite-dimensional optimization problems can be found, e.g., in [1]}. We first remark that all function \(J\)  is a convex function. Moreover, it’s Gâteaux-derivative \(J^{\prime }\)  exists, which means that for all \(\eta \in \mathcal {U}\) , there exists \(J^{\prime } (h)\in \mathcal {U}\)  such that for all \(\eta \in \mathcal {U}\) ,
\(\langle J^{\prime }(h),\eta \rangle =\lim _{t\rightarrow 0}\frac{J(h+t\eta )-J(h)}{t}.\) 
